You called?
Sure, but that's just a matter of how the terminology is used. Setting aside natural logs for a moment, consider that log10(1000) is 3, where 3 is simultaneously the "exponent" to which 10 must be raised to equal 1000, and the "logarithm," in the sense that it's the result of the log10 function, as might be found by looking in a table of logarithms.
I understand what you mean, but the problem is that the terminology has a precisely defined meaning, and if you ignore the convention you will be misunderstood. It would be like saying that a certain O(log n) algorithm has an exponential complexity -- the meaning is completely different. The growth of a function concerns the dependent variable, so y=e^x has exponential growth, while y=ln(x) is logarithmic. See also:
http://en.wikipedia.org/wiki/Logarithmic_growth
http://en.wikipedia.org/wiki/Exponential_growth
(Sorry for the off-topic tangent!)
exp(x) takes e (the constant) and raises it to the x power. Log(x) finds the power to raise the base (usually 10 but sometimes e) to produce x. They are essentially inverses of each other. Log(exp(x)) = x;
Graphically they are quite different: http://www.wolframalpha.com/input/?i=exp%28x%29
http://www.wolframalpha.com/input/?i=ln%28x%29